Biomathematics: Mathematics of Biostructures and Biodynamics
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Table of contents
- Cover
- Contentsv
- Chapter 1. Introduction1
- References4
- Chapter 2. Counting, Algebra and Periodicity - the Roots of Mathematics are the Roots of Life7
- 2.1 Counting and Sine7
- 2.2 Three Dimensions; Planes and Surfaces, and Surface Growth9
- 2.3 The Growth of Nodal Surfaces - Molecules and Cubosomes16
- References 226
- Chapter 3. Nodal Surfaces of Tetragonal and Hexagonal Symmetry, and Rods27
- 3.1 Non Cubic Surfaces27
- 3.2 Tetragonal Nodal Surfaces and their Rod Structures27
- 3.3 Hexagonal Nodal Surfaces and their Rod Structures36
- References 345
- Chapter 4. Nodal Surfaces, Planes, Rods and Transformations47
- 4.1 Cubic Nodal Surfaces47
- 4.2 Nodal Surfaces and Planes50
- 4.3 Cubic Nodal Surfaces and Parallel Rods56
- 4.4 Transformations of Nodal Surfaces68
- References 472
- Chapter 5. Motion in Biology73
- 5.1 Background and Essential Functions73
- 5.2 The Control of Shape - the Natural Exponential or cosh in 3D76
- 5.3 The Gauss Distribution (GD) Function and Simple Motion81
- 5.4 More Motion in 3D93
- References 5102
- Chapter 6. Periodicity in Biology - Periodic Motion105
- 6.1 The Hermite Function105
- 6.2 Flagella- Snake and Screw Motion111
- 6.3 Periodic Motion with Particles in 2D or 3D116
- 6.4 Periodic Motion with Rotation of Particles in 2D127
- References 6130
- Chapter 7. Finite Periodicity and the Cubosomes131
- 7.1 Periodicity and the Hermite Function131
- 7.2 Cubosomes and the Circular Functions133
- 7.3 Cubosomes and the GD-Function - Finite Periodicity and Symmetry P139
- 7.4 Cubosomes and the GD-Function - Symmetry G143
- 7.5 Cubosomes and the GD Function - Symmetry D147
- 7.6 Cubosomes and the Handmade Function152
- References 7162
- Chapter 8. Cubic Cell Membrane Systems/Cell Organelles and Periodically Curved Single Membranes163
- 8.0 Introduction163
- 8.1 Cubic Membranes163
- 8.2 The Endoplasmatic Reticulum169
- 8.3 Protein Crystallisation in Cubic Lipid Bilayer Phases and Cubosomes - Colloidal Dispersions of C175
- 8.4 From a Minimal Surface Description to a Standing Wave Dynamic Model of Cubic Membranes177
- 8.5 Periodical Curvature in Single Membranes183
- References 8190
- Chapter 9. Cells and their Division - Motion in Muscles and in DNA193
- 9.1 The Roots and Simple Cell Division193
- 9.2 Cell Division with Double Membranes201
- 9.3 Motion in Muscle Cells206
- 9.4 RNA and DNA Modelling213
- References 9220
- Chapter 10. Concentration Gradients, Filaments, Motor Proteins and again- Flagella223
- 10.1 Background and Essential Functions223
- 10.2 Filaments227
- 10.3 Microtubulus and Axonemes235
- 10.4 Motor Proteins and the Power Stroke244
- 10.5 Algebraic Roots Give Curvature to Flagella247
- References 10255
- Chapter 11. Transportation257
- 11.1 Background - Examples of Docking and Budding with Single Plane Layers, and Other Simple Example257
- 11.2 Docking and Budding with Curved Single Layers265
- 11.3 Transport Through Double Layers273
- References 11284
- Chapter 12. Icosahedral Symmetry, Clathrin Structures, Spikes, Axons, the Tree, and Solitary Waves285
- 12.1 The icosahedral symmetry285
- 12.2 Hyperbolic Polyhedra, Long Cones, Cylinders and Catenoids294
- 12.3 Cylinder Division and Cylinder Fusion - Cylinder Growth299
- 12.4 Solitary Waves, Solitons and Finite Periodicity305
- References 12311
- Chapter 13. Axon Membranes and Synapses - A Role of Lipid Bilayer Structure in Nerve Signals313
- 13.1 The Nerve Impulse313
- 13.2 At the Action Potential Region of the Membrane there is a Phase Transition in the Lipid Bilayer315
- 13.3 A Model of a Phase-Transition/Electric Signal Coupling at Depolarisation and its Physiological317
- 13.4 Transmission of the Nerve Signal at the Terminal Membrane of the Neurons - Synaptic Transmissio327
- 13.5 Synchronisation of Muscle Cell Activation333
- 13.6 The General Anaesthetic Effect335
- 13.7 Physiological Significance of Involvement of a Lipid Bilayer Phase Transition in Nerve Signal C337
- References 13338
- Chapter 14. The Lung Surface Structure and Respiration341
- 14.1 The Alveolar Surface341
- 14.2 Lung Surfactant342
- 14.3 Structure of Tubular Myelin - A Bilayer arranged as the Classical CLP-Surface344
- 14.4 The Existence of a Coherent Surface Phase Lining the Alveoli349
- 14.5 Respiration357
- 14.6 Physiological Significance of the Existence of an Organised Surface Phase at the Alveolar Surfa359
- References 14361
- Chapter 15. Epilogue363
- Acknowledgement372
- References 15372
- Appendix 1. The Plane, the Cylinder and the Sphere375
- Appendix 2. Periodicity385
- Appendix 3. The Exponential Scale, the GD function, Cylinder and Sphere Fusion399
- Appendix 4. The Exponential Scale, the Planes and the Natural Function, Addition and Subtraction 409409
- Appendix 5. Multiplication of Planes, Saddles and Spirals419
- Appendix 6. Symmetry431
- Appendix 7. The Complex Exponential, the Natural Exponential and the GD- Exponential - General Examp447
- Appendix 8. Classical Differential Geometry and the Exponential Scale463
- Appendix 9. Mathematica (Contains the Mathematica scripts used for calculating the equations for the477
- Subject Index521
This book presents new mathematics for the description of structure and dynamics in molecular and cellular biology. On an exponential scale it is possible to combine functions describing inner organisation, including finite periodicity, with functions for outside morphology into a complete definition of structure. This mathematics is particularly fruitful to apply at molecular and atomic distances. The structure descriptions can then be related to atomic and molecular forces and provide information on structural mechanisms. The calculations have been focussed on lipid membranes forming the surface layers of cell organelles. Calculated surfaces represent the mid-surface of the lipid bilayer. Membrane dynamics such as vesicle transport are described in this new language. Periodic membrane assemblies exhibit conformations based on the standing wave oscillations of the bilayer, considered to reflect the true dynamic nature of periodic membrane structures. As an illustration the structure of an endoplasmatic reticulum has been calculated. The transformation of such cell membrane assemblies into cubosomes seems to reflect a transition into vegetative states. The organisation of the lipid bilayer of nerve cells is analyzed, taking into account an earlier observed lipid bilayer phase transition associated with the depolarisation of the membrane. Evidence is given for a new structure of the alveolar surface, relating the mathematical surface defining the bilayer organisation to new experimental data. The surface layer is proposed to consist of a coherent phase, consisting of a lipid-protein bilayer curved according to a classical surface - the CLP surface. Without employing this new mathematics it would not be possible to give an analytical description of this structure and its deformation during the respiration cycle. In more general terms this mathematics is applied to the description of the structure and dynamic properties of motor proteins, cytoskeleton proteins, and RNA/DNA. On a macroscopic scale the motions of cilia, sperm and flagella are modelled.
This mathematical description of biological structure and dynamics, biomathematics, also provides significant new information in order to understand the mechanisms governing shape of living organisms.
This mathematical description of biological structure and dynamics, biomathematics, also provides significant new information in order to understand the mechanisms governing shape of living organisms.